One of the main volcanic processes affecting road bridges
are lahars, which are flows of water and volcanic material running down the
slopes of a volcano and river valleys. Several studies have evidenced the
effects of other volcanic processes on road infrastructure; however,
limited information is available about the effects of lahars on bridges.
In this paper, bridge failure models due to lahars are proposed and, based
on these, fragility curves are developed. Failure models consider the limit
state of pier and abutment overturning, and deck sliding caused by lahars.
Existing physical models are used to stochastically characterize lahar loads
and overturning momentum on bridges. Monte Carlo simulations are applied to
quantify the probability of bridge failure given by different lahar depths.
Fragility curves of bridges are finally parameterized by maximum likelihood
estimation, assuming a cumulative log-normal distribution. Bridge failure
models are empirically evaluated using data on 15 bridges that were affected
by lahars in the last 50 years. Developed models suggest that decks fail
mainly due to pier and/or abutment overturning, rather than deck-sliding
forces. Moreover, it is concluded that bridges with piers are more
vulnerable to lahars than bridges without piers. Further research is being
conducted to develop an application tool to simulate the effects of expected
lahars on exposed bridges of a road network.

Lava and pyroclastic flows may destroy road infrastructure, but the
probability of occurrence of these events is low and exposed areas are
limited (Wilson et al., 2014). Considering that risk is a function of the
hazard, exposure and vulnerability (UNISDR, 2009), a lower risk of lava and
pyroclastic flows on road infrastructure is consequently expected. Lahars
are flows of water, rock fragments and debris that descend from the slopes
of volcanoes and river valleys. Road infrastructure reached by lahars are
damaged physically and operationally (Smith and Fritz, 1989). Volcanic
debris and sediments transported by lahars make these flows especially
destructive. Lahar flows also scour the riverbed, permanently affecting in
the foundations of the exposed infrastructure (Vallance and Iverson, 2015;
Muñoz-Salinas et al., 2007; Nairn, 2002). Wilson et al. (2014)
demonstrated that bridges and culverts are the road infrastructure elements
most exposed and vulnerable to lahars. Blong (1984) and Wilson et al. (2014)
reported that 300 km of roads were damaged and 48 bridges were affected
because of Mount St Helens (USA) eruption in 1980. Moreover, the eruption
of Villarrica and Calbuco volcanoes, which occurred in Chile in 2015,
collapsed four of six bridges reached by lahars (MOP, 2015a, b).

Fragility curves are commonly integrated in available risk modelling tools.
For example, in the United States, the Federal Emergency Management Agency
(FEMA) developed HAZUS-MH tool for risk management of structures and
infrastructure. This GIS-based software covers three natural hazards,
earthquakes, floods and hurricanes, excluding the volcanic hazard from the
analysis (FEMA, 2011). Likewise, the RiskScape software developed by the
National Institute of Water and Atmospheric Research (NIWA) of New Zealand
included the effects of earthquakes, tsunamis, floods, hurricanes and
volcanic eruptions on assets such as buildings, roads and power lines.
Nevertheless, the effects of volcanoes are only accounted for in terms of
ash fall and the temporary interruption of infrastructure operation (Kaye,
2008). Fragility curves have been developed for some infrastructure and
utilities exposed to volcanic hazard, such as buildings and electric
transmission systems (Spence et al., 2005, 2007; Jenkins and
Spence, 2009; Zuccaro and De Gregorio, 2013). In particular, Wilson et al. (2017) developed fragility curves for road infrastructure exposed to tephra
fall. However, fragility curves for road infrastructure exposed to lahars
have not been developed regardless of empirical evidence on their destructive
effects (Wilson et al., 2014).

From the available literature and current state of practice, it is
concluded that no bridge fragility curves exposed to lahar flows have been
developed. To characterize bridge fragility to lahars, failure probability
of primary structural elements is required, namely substructure (i.e. piers
and abutments) and deck. Piers are columns designed to act as interior
support for a bridge superstructure; abutments are the end support for a
bridge superstructure; and deck is the component that supports wheel loads
directly and is supported by other components (AASHTO, 2012).

The main objective of this study is to propose simplified bridge failure
models and fragility curves by considering pier and abutment overturning, as
well as deck sliding caused by lahars. The research starts with the
characterization of the lahar process and the physical effects on bridges.
From this analysis lahar depth was identified as a critical stochastic
variable that is representative of the hazard intensity. Failure models are then
proposed, considering the limit state of pier and abutment overturning due
to lahar demanding forces and reduced supply moment caused by scour. In the
case of bridge deck, the limit state is analysed by considering lahar
tangential force and supplied deck friction. Monte Carlo simulations are
applied to estimate the bridge failure probability by considering different
lahar depths, allowing the fragility curves to be calibrated. The analysis is
performed by considering one-span and multiple-span bridges.

Best-fit probability functions are finally proposed by considering cumulative
log-normal distribution and their corresponding parameters from maximum
likelihood analysis. Limited historical data are available to empirically
validate the proposed fragility curves, but models were compared with
post-event data from 15 bridges, being in all cases consistent with developed
models. Future research should be conducted to statistically validate
developed fragility curves with reliable empirical data.

2.1 Physical description of lahar flows

Lahars are high-velocity flow containing a mix of volcanic debris and water,
travelling through ravines and riverbeds (Pierson et al., 2009). Lahar flows
originated from an abrupt melting of snow and/or ice caused by the heat
flow derived from lavas or pyroclastic flows issued during a volcanic event
or by avalanches of non-consolidated volcanic material during intense
rainfall or rupture of a lake or pond (Waitt, 2013). Lahars are categorized according
to their sediment ∕ water ratio into debris flows and Hyperconcentrated flows
(Smith and Fritz, 1989). Debris flows are highly viscous slurries of
sediment and water. Debris flows are capable of transporting gravel-sized
debris in suspension, and their concentration of solid particles ranges
between 75 % and 80 % in weight or 55 % and 60 % in volume.
Hyperconcentrated flows have high-suspended fine contents, predominantly
due to fluid motion and properties. The solid concentrations of
Hyperconcentrated flows can represent up to 55 % to 60 % of the total
weight and 35 % to 40 % of the total volume (Pierson et al., 2009). Thus,
the variability of the lahar density and rheology should be considered in
the development of a risk model of this hazard.

The flow of lahars is guided by gravity and is capable of impacting elements
located tens of kilometres away from the crater of the volcano (Parfitt and
Wilson, 2008). Furthermore, lahars can reach velocities up to
140 km h−1 as observed in Mount St
Helens in the United States in 1980 (Pierson, 1985). The velocity and
composition of lahars make them highly destructive.

According to Vallance and Iverson (2015) and Bono (2014), the most important
processes of a lahar are the erosion of the steep slopes and the scouring of
beds and fluvial terraces. Even more significant is the erosion observed in
steeper river valleys with weaker beds. Watery sediment floods are more
erosive than sediment-rich flows. The scour of the riverbed drags material
blocks, revetment and vegetation. Thus, most of the bridges affected by
lahars are located in valleys in volcanic areas. The erosion and the
associated loads of high velocity lahars, and the impact of debris
travelling with them, may cause the collapse or permanent deterioration of
bridges (Nairn, 2002). This explains, in part, the high vulnerability of
bridges to lahar flows.

Relevant drivers of the destructive potential of a lahar affecting a bridge
are the bed material, the slope, the season in which the lahar occurs, the
existence of a glacier, rainfall and the prevailing temperatures during
winter. The destructive potential of a lahar increases when the eruption
occurs at the end of the winter, since in this season there is more
accumulated snow compacted in layers and more volume of ice melting. This
condition is enhanced if winter temperatures are low, because greater
volumes of ice and snow melting in shorter lapses of time may increase the
intensity of lahars (Moreno, 2015).

2.2 Bridge fragility curves for lahar risk modelling

In order to incorporate the uncertainty of the characteristics of lahar
flows and the bridge engineering design (X), the use of fragility curves
to quantify the probability of bridge failure due to lahars is proposed.
Fragility curves express the probability that the damage state (DS) of a
system exceeds different levels (dsi= slight, moderate, extensive
or complete), given a certain hazard intensity (IM) (See Eq. 1). The
fragility curves allow the failure probability of a system to be quantified due
to an event of a specific intensity (Rossetto et al., 2013), representing
the system vulnerability to a natural hazard. In this study, bridge
fragility curves for a complete damage state level are developed.

(1)P(DS≥dsiIM)

Schulz et al. (2010) define four approaches for developing a system's
fragility curves. First, there is the empirical approach, which is based on
historical data and/or experiments. Second, fragility curves can be based on
expert opinions as well. Third, fragility curves can also be developed
using an analytical approach through models that characterize the limit
state of the element, based on probabilistic and deterministic variables
defining the system. Finally, a hybrid method can be used, which combines two or more of
the approaches described above.

Since there are no existing models addressing lahar risk on bridges, a
challenge for the development of bridge fragility curves consists of
defining a unified lahar hazard intensity (IM). In general, the flow depth
is a measure of hazard intensity of natural events that involve liquid
flows. In the flood module of the HAZUS-MH software, the Federal Emergency
Management Agency developed fragility curves using flow depth to
quantify the hazard intensity (FEMA, 2011). Tsubaki et al. (2016) use the
same variable (flow depth) for measuring the flood intensity when developing
embankment fragility curves. Wilson et al. (2014) propose flow depth as
one of the potential intensity measures for developing fragility curves
related to lahar flows. In this paper lahar depth was used as the
lahar hazard intensity, because this variable is correlated to
other lahar flow characteristics, such as velocity and scour demand (Arneson
et al., 2012).

3 Development of failure models for bridge pier/abutment overturning and deck sliding due to lahars

3.1 Conceptual model

In order to model bridge fragility due to lahars, the analytical approach is
based on reliability principles. The assessment of the bridge
reliability can be considered a supply and demand problem associated with a
bridge–lahar system defined by its basic variables vector (X). The supply
function (S(X)) of the bridge corresponds to its capacity to resist the
loads of the lahar. It is directly related to the design of the structural
element. The demand function (D(X)) represents the load applied by the
lahar on the bridge. The limit state function (g(X)) of the bridge–lahar
system is given by the difference between the supply and demand functions
(gX=DX-S(X)). If g(X) is lower than zero,
the lahar loads on the structure are greater than the bridge capacity and
hence, the bridge will fail.

With the purpose of conceptualizing the loads applied by the lahar flow on
the bridge components, a bridge–lahar model was developed, which is shown in
the free-body diagram in Fig. 1. It shows the generic cross section of a
bridge and the main physical loads applied by the lahar on the bridge. The
cross section of the bridge in Fig. 1 is composed by the substructure
(pier/abutment) and the superstructure (deck and beams). The proposed
failure models can be adapted to different bridge design criteria. In this
paper, the Chilean design standards are considered for the fragility curve
calibration. Thus, the proposed models assume that the foundation has no
piles. This assumption is based on the fact that 88 % of the bridges
exposed to the volcanic hazard from the Villarrica and Calbuco volcanoes do
not have piles (Moreno, 1999, 2000). Additionally, it assumes a
simple support of the superstructure on the piers and abutments, because
this is the most common support mechanism in Chilean bridges.

Figure 1Free-body diagram of bridge resisting and demanding forces and
moments in the presence of a lahar.

Figure 1 shows a lahar with depth hLahar acting on a bridge of width T.
Each pier or abutment of the bridge has a weight W. The foundation of the
bridge's substructure (pier/abutment) was designed with a depth
Ys, which
represents the supply or capacity of the bridge to resist scour. The lahar
flow demands a scour Yd on the bed around the foundation. The modelled
lahar generates a hydrodynamic pressure pw, which acts perpendicular to
the bridge. This pressure produces a resulting hydrodynamic tangential force
Fwi on the piers and abutments, and a force Fws on the bridge
superstructure. Furthermore, the debris transported by the lahar colliding
with the bridge impacts the structure with a force Fi. The tangential
force Ft corresponds to the sum of the hydrodynamic force and the
debris impact force applied to the superstructure. The deck of the bridge
resists the sliding caused by the lahar tangential force Ft with a
friction force Fr. All the forces applied to the substructure produce a
net resulting moment Mn on the lower-right vortex of the
foundation. The net moment Mn is equal to the difference between the
overturning moment (Mv) generated by the hydrodynamic force (Fwi)
and the debris impact (Fi), and the resistant moment (Mr) produced
by the weight W of the bridge.

3.2 Bridge failure mechanisms due to lahars

The hydrodynamic pressure of the lahar flow (pw) and the impact force
of the debris (Fi) can cause the overturning of bridge piers and
abutments. This is further enhanced by the scour that these flows generate
around the foundations. The hydrodynamic pressure of the lahars, together
with the potential impact of debris, can cause deck sliding.

With the aim of analysing the effects of lahars on bridges, failure
mechanisms associated with three bridge components are defined: pier
overturning, abutment overturning and sliding of the bridge superstructure.
In addition to these failure mechanisms, the access embankment of the bridge
may collapse. However, this component is not included in the modelling due
to its lower replacement cost in relation to other bridge components. All
these failure mechanisms are consistent with the postulates of Wilson et al. (2014) and the records of the lahar effects as a result of the eruptions of
the Villarrica volcano and the Calbuco volcano in 2015 (MOP, 2015a, b). Images in Fig. 2a and b show the Río Blanco Bridge (Chile)
before and after a lahar flow following the eruption of Calbuco volcano in
2015. Figure 2 shows the structural collapse of the bridge due to the
overturning of the pier and subsequent sliding of the deck.

3.2.1 Substructure overturning (piers and abutments)

Both piers and abutments are components susceptible to overturning due to
lahars. These dense and fast-travelling flows generate a resulting
hydrodynamic force (Fwi) on the bridge substructure, which entails an
overturning moment (Mwi). The impact force (Fi) of the debris on
piers and abutments produces the overturning moment (Mi). The bridge
weight W generates a moment (Mr) resisting the substructure
overturning.

Through equilibrium of moments, considering the turning point O located in
the vertex of the foundation, it is possible to evaluate the stability of
the bridge piers and abutments in the presence of a lahar flow of a specific
intensity. The overturning of piers and abutments is produced if the
overturning moment (Mv=Mwi+Mi) caused by the lahar on the
component is greater than the resistant moment (Mr). In other words,
the overturning is produced when the net moment (Mn) is less than zero.

A lahar can also cause the overturning of piers and abutments when the depth
of the scour generated by the flow on the bed Yd(X) is greater than the
design scour of the substructure Ys(X).

The above allows us to establish the limit state function gSO(X) related
to the overturning of piers and abutments due to lahars. This function
allows the overturning probability of the substructure to be quantified
considering the parameters (X) of the system and the lahar intensity
hLahar:

(2)PSO=PgSOX≤0(3)gSOX=minMrX-MvX;YsX-YdX.

This function indicates that, given a lahar with height hLahar, the
substructure will overturn if the overturning moment Mv is greater than
the resistant moment Mr and/or the lahar scour demand Yd is higher
than the design scour of the bridge Ys.

The scour caused by lahar flows near the foundations contributes to a
greater vulnerability of these bridge components, since the lahars produce
destabilization and weakening around the foundation of piers and abutments.
If there is scour in the bed, the foundation of the pier or abutment will be
exposed to a higher hydrodynamic pressure. This load is higher in the case
of lahars, given their greater density and velocity in relation to normal
floods. A greater scour demand will imply a larger surface affected by the
hydrodynamic pressure. In turn, this means a greater resulting hydrodynamic
force (Fwi) and, therefore, a greater moment associated with this force
(Mwi).

3.2.2 Deck sliding

In the case where the lahar height exceeds the bridge clearance, the lahar
flow will exert a hydrodynamic pressure on the bridge superstructure. There
is also the possibility that the debris transported by the lahar flow
impacts the bridge deck. This debris impact force (Fis), together with
the hydrodynamic force (Fws), can cause failure due to deck sliding. The
presence of microscopic imperfections between the contact surfaces of the
superstructure (beams) and the substructure (piers and abutments) produces a
static friction force (Fr) that opposes the start of the deck sliding.

Through the equilibrium of forces it can be inferred that the deck of a
bridge subjected to a lahar will slide if the resulting tangential force
(Ft=Fws+Fis) is higher than the static friction force (Fr)
between the substructure and the superstructure. It should be highlighted
that this force is zero if the lahar height is lower than the bridge
clearance.

This allows the limit state function gDS(X) to be established. It is
associated with the superstructure failure due to its potential sliding:

(4)PDS=PgDSX≤0,(5)gDSX=FrX-FtX.

The limit state function defined in Eq. (5) implies that, under attribute
X, if the friction force is lower than the tangential force produced by
the lahar, the failure mechanism associated with sliding will be activated.

4 Proposal for modelling substructure overturning and deck sliding due to lahars

4.1 Physical models to estimate limit state functions

Once the limit state functions have been analytically defined, the loads
presented in the free-body diagram have to be quantified. Therefore,
physical existing models are used and integrated.

4.1.1 Lahar hydraulic attributes

First, the lahar mean velocity (vLahar) is quantified with Eq. (6),
suggested by Chen (1983, 1985) for a fully dynamic debris flow in a channel
with an arbitrary geometric shape. In this study, a rectangular shape is
assumed. This formula incorporates the rheology of the lahar through the
consistency index (μLahar), which was quantified by Laenen and
Hansen (1988) for the case of lahars. The slope of the bed around the bridge
(i) was measured by Bono (2014) to model the hydraulic behaviour of the
main lahars of the Villarrica volcano. According to the measurements of Bono
(2014), the range of the riverbed slope is limited, so there is a strong
relationship between lahar depth and velocity.

(6)vLahar=25γLaharμLahar12i1/2ALaharPLahar3/2

The lahar hydrodynamic pressure (pw) is estimated with the AASHTO model (2012). This model considers a triangular distribution of this pressure,
taking a value of zero in the deepest point and a maximum value in the flow
surface. The hydrodynamic pressure is a function of the specific weight of
the flow (γLahar), its velocity (vLahar) and the drag
coefficient (CD).

(7)pw,max=CDγLahargvLahar2

4.1.2 Scour models

The lahar scour demand is based on the empirical equation proposed by
Arneson et al. (2012). Müller (1996) compared 22 equations proposed in
the literature to estimate scour; he used empirical data from 384 field
measurements of 56 bridges. The conclusion of Müller (1996) was that the
equation proposed by Arneson et al. (2012) in the Hydraulic Engineering
Circular No. 18 (HEC-18) was suitable for quantifying the scour depth.

Debris transported by the flows accumulates in the bridge piers, creating an
additional obstruction to the flow. To incorporate the debris accumulation,
the scour demand on the piers (Yd−p) is modelled with Eqs. (8) and (9)
of the NCHRP (2010). The equations proposed by the NCHRP adjust the scour
model proposed by the HEC-18 to estimate the scour generated by debris flows
and lahars. The adjusted model considers a triangular or rectangular debris
accumulation (shape factor KE) with height Hd and width Wd to
estimate an effective widening (bd*) of the pier with width b
(See Eq. 9). It should be noted that factors K1, K2 and
K3 are correction factors of the pier shape, the flow angle and the bed
condition, respectively.

According to the HEC-18, the scour demand on the abutments (Yd−a) is
based on the flow depth (hLahar), the flow width (bFlow), the
bridge length (LBridge) and a bed condition amplification factor
(α).

(10)Yd-a=αhLaharbFlowLBridge6/7-hLahar

The scour supply is estimated with models adapted from bridge design
manuals. For example, Breusers et al. (1977) stipulate that Eqs. (11)
and (12) assess the design scour of piers (Ys−p) and abutments
(Ys−a). These equations include variables such as design height
(hDesign), pier width (b) and correction factors by flow angle, pier
shape, among others. For the design of bridge foundations, the Highway
Manual of the Ministry of Public Works (MOP, 2016) suggests considering a
depth equal to 2.0 m below the scour level. The design height of the
flow (hDesign) is considered equal to the bridge vertical clearance
minus the distance between the design water depth and the lowest level of
the superstructure, which is equal to 1.0 m according to the Highway
Manual (MOP, 2016).

4.1.3 Substructure overturning moment and deck tangential force

The overturning moment (Mv) produced by lahars on the bridge
substructure is given by the sum of the hydrodynamic moment (Mwi) and
the debris impact moment (Mi). The tangential force (Ft) on the
deck corresponds to the sum of the resulting force from the hydrodynamic
pressure on the deck (Fws) and the debris impact force (Fis).
Considering the pressure model showed in Eq. (7), the hydrodynamic moment
generated by the lahar on the substructure (Mwi) can be estimated. In
the case of substructure, the hydrodynamic moment is separated into two
parts: the foundation and the column. This separation is supported by the
fact that these elements have different geometry and that the pressure has a
triangular distribution over the foundation and trapezoidal distribution
over the column (Fig. 1).

Mwi=Mw,found+Mw,column(13)=Fw,foundyw,found+Fw,columnyw,column

The resulting hydrodynamic force exerted by the lahar on the foundation
(Fw, found) and the height at which this force acts with respect to the
turning axis (yw, found) are given by Eqs. (14) and (15), where the
variable T corresponds to the bridge width:

(14)Fw,found=TCDγLahar2gvLahar2Yd2hLahar+Yd(15)yw,found=Ys-Yd3.

The hydrodynamic force on the column (Fw, column) and its application
point (yw, column) depend on whether the height of the lahar exceeds the
height of the column or not. To incorporate this, the variable h*
was defined, which is given by the minimum between the lahar height
(hLahar) and the column height (hDesign).

In order to quantify the hydrodynamic force of the lahar on the deck
(Fws), three criteria should be met: (1) the lahar height is lower
than the bridge clearance, (2) the lahar height is greater than the
clearance but lower than the roadway level, (3) the lahar height is greater
than the roadway level. In the model, the roadway level is given by the sum
of the substructure height (hDesign), and the superstructure thickness
(eSuper).

To quantify the impact of debris on the bridge, the model of Haehnel and
Daly (2004) is used. This model assesses the impact force through a
one-degree-of-freedom system assuming a rigid structure. Thus, the impact
force of gravel transported by a lahar on the bridge is based on the flow
velocity (vLahar), the specific weight of the gravel (γGravel), the gravel diameter (DGravel) and the contact stiffness of
collision (k^). Debris impact force on the deck (Fis) is given
by Eq. (19).

The moment of debris impact (Mi) on the substructure with respect to
the rotation axis is shown in Eq. (20). This indicates that if the impact
height (himp) is greater than the substructure (hDesign), the
associated moment is zero. For the impact height, a triangular distribution
with the mode equal to the lahar height is assumed, considering that the
debris tends to collect in the flow surface (Zevenbergen et al., 2007).

Mi=(20)vLaharγGravel43πDGravel23himp+Yshimp≤hDesign0himp>hDesign

4.1.4 Substructure resistant moment and deck friction force

The substructure capacity to oppose overturning depends on the design and condition of the bridge
elements, including the bridge geometry, materials and
the scour design (Ys−p and Ys−a). Thus, the lahar loads on the
bridge and the scour are considered only in the demand function (overturning
moment Mv). The resistant moment (Mr) of the substructure to
lahars is given by the weight (W) of the pier or abutment and the elements
that are supported on it. Among the elements supported by the substructure,
the superstructure and the soil on the abutment foundations must be
considered. The weight of the piers and abutments without considering the
soil and the superstructure are as follows:

(21)WSub=γSubYsT2+γSubhDesignbT.

The weight of the soil on the abutment foundation in the access to the
bridge is given by Eq. (22).

(22)WSoil-Abutment=0.5γSoilhDesign(T2-bT)

The model considers that the weight of the superstructure is distributed
uniformly in all its supports (NA). Thus, the force exerted by the
superstructure on each foundation is as follows:

(23)WSuper=γSuperTLBridgeeSuperNA.

Since the elements of the modelled bridge are symmetrical with respect to the
vertical axis, the weight acts at a distance T∕2 from the overturning
point. Thus, the resistant moment of the substructure is given by the
following expression:

(24)Mr=(WSub+WSoil-Abutment+WSuper)T2.

Finally, the force that opposes the deck sliding corresponds to the friction
between the superstructure and the substructure. This force is given by
Eq. (25):

(25)Fr=μsNSuper=μsuperγSuperTLBridgeeSuper.

4.2 Values of the variables involved in the limit state functions

In order to quantify the independent variables of the limit state function,
the first step is to define the nature of the variables, based on their
degree of uncertainty. The system bridge–lahar presents random variables
associated with lahar hazard, such as lahar density and debris accumulation.
To quantify these variables, probability distribution functions are used,
based on studies prepared by the Chilean National Geology and Mining Service
(Sernageomin) (Castruccio et al., 2010; Bono, 2014) and the United States
Geological Survey (Pierson et al., 2009; Vallance and Iverson, 2015).

Furthermore, regarding variables associated with the bridges' capacity to
resist lahars, random variables are also considered due to the uncertainty
in the bridge design. Goodness of fit tests were undertaken to determine the
probability functions and the parameters of these variables using the
information from the Chilean bridge inventory and the Highway Manual of the
Ministry of Public Works (MOP, 2016). Table 1 summarizes the values of the
variables involved in the limit state functions.

5.1 Monte Carlo simulations for fragility curve calibration

Reliability analysis comprises analytical solution methods and numerical
solution methods. Analytical solution methods feature the first-order
second-moment (FOSM) method, the first-order reliability method (FORM) and
the second-order reliability method (SORM). Numerical solution methods
include the Monte Carlo simulation (MCS) and the response surface
method (RSM). The MCS method is used to develop bridge fragility curves due
to lahars. The choice of the MCS as the solution method is based on the
limitations of the analytical solution methods with regard to the
probability distributions of the random variables (Schultz et al., 2010).
MCS allows us to incorporate the uncertainty of the characteristics of lahars
and the structure in the quantification of the bridge failure probability,
without the mentioned limitation.

With the limit state functions and variables already defined, the Monte
Carlo simulations can be performed. Therefore, a fixed intensity lahar
h1 is considered. The probability distributions of the system's random
variables imply the obtainment of different values of limit state functions
g(X). If this function is less than zero in a specific simulation, it
means that in this simulation the bridge fails due to a lahar with intensity
h1. The bridge failure probability due to a lahar of intensity h1
is equal to the sum of the number of simulations where function g(X) is
negative, divided by the number of total simulations with this intensity
(NS) (Vorogushyn et al., 2009).

(26)PFailure=PgX<0|H=h1=∑i=1NSkiNS,(27)ki=1giX<00giX≥0,

Simulations with fixed intensity h1 allow the failure
probability of the fragility curve at the abscissa h1 to be quantified. This experiment
is carried out repeatedly for several intensity levels, to obtain the
complete fragility curve for each failure mechanism identified.
Specifically, 10 000 simulations were performed for each intensity level.
The failure probability is quantified for lahar heights discretized every
0.25 m.

5.2 Calibrated bridge fragility curves due to lahars

5.2.1 Fragility curves by bridge failure mechanism

Once the supply and demand functions of the failure mechanisms are defined
together with their variables, simulations are run for a fixed lahar height
level h1. The percentage of simulations where function gSO(X) is
less than zero is equivalent to the overturning probability of the
substructure in the presence of a lahar of h1. After doing this for
different lahar height levels, the overturning fragility curves of the piers
and abutments are obtained. The same experiment was performed for the
function gDS(X) to calibrate the deck-sliding fragility curve. Figure 3
shows the fragility curves by bridge failure mechanism.

Figure 3Fragility curves for bridge substructure overturning and deck
sliding due to lahars.

The analysis of substructure overturning fragility curves allows us to
conclude that, when impacted by lahar flows, piers are more susceptible to
overturn than the abutments. Given any intensity level of the hazard, piers
have a greater probability of overturning than abutments. The functional
shape of the overturning fragility curves shows that, regarding the
abutments; the maximum failure probability increase is achieved when the
intensity grows from 2.5 to 2.75 m, where the failure probability increases
by 41.8 percentage points. In the case of piers, the maximum growth of the
probability of failure is reached between 1.75 and 2.0 m increasing the
overturning probability by 37.4 percentage points.

When analysing the deck-sliding fragility curve, the deck failure
probability is zero if the lahar intensity is less than or equal to 2.50 m. This
is mainly due to the fact that a low-height lahar does not reach the bridge
clearance and, consequently, the flow does not affect the superstructure.
Nevertheless, if the intensity of the lahar exceeds this level, the failure
probability increases rapidly. The growth rate of this fragility curve also
has a maximum, which is reached when the lahar arrives at 3.25 m,
particularly if the lahar increases from 3.0 to 3.25 m and the sliding
probability of the deck increases by 45.5 percentage points. This is mainly due
to the fact that if the lahar reaches 3.50 m, it already touches the road
elevation of most bridges of the inventory.

5.2.2 Fragility curves by bridge categories

The previous analysis allows us to conclude that a relevant factor in a
bridge failure due to a lahar is the presence of piers. Therefore, two
bridge categories were defined: bridges with one span (C1) and bridges with
multiple spans (C2). Category C1 corresponds to bridges with substructure
composed only of abutments and category C2 represents bridges with one or
more piers.

To obtain the fragility curves for these two bridge categories, each
simulation considered that the failure of the bridge occurs when at least
one of its components fails. For example, a bridge of category C1 fails when
the abutment overturns and/or when the deck slides. A category C2 bridge
fails when the pier or abutment overturns and/or the deck slides. The
fragility curves of C1 and C2 bridges are developed using the same failure
models for the abutments and the deck. Figure 4 shows the fragility curves
for both bridge categories, in addition to the failure probability of each
component in a histogram.

Figure 4 allows us to conclude that bridges with one span (C1) are stronger
than bridges with two or more spans (C2) in the presence of lahar flows. The
reason is that piers are more susceptible to overturn than abutments. The
failure of bridges with one span is guided by the abutments overturning,
while in the bridges with multiple spans, the failure is guided by the piers
overturning. The deck sliding is not a triggering factor of bridge failures
due to lahars generated by the Villarrica and Calbuco volcanoes.

5.3 Parameterization of bridge fragility curves due to lahars

When considering risk management from a strategic point of view, the
parameterization of bridge fragility curves due to lahars entails a series
of advantages. It allows us to directly estimate the failure or collapse
probability of each bridge category based on the lahar depth. Moreover, it
allows the failure probability to be continuously quantified, that is, not every
25 cm of lahar.

For the parameterization of fragility curves, a cumulative log-normal
distribution is considered. When assessing parameters μ and β of
the cumulative log-normal distribution reflecting the fragility curve, the
bridge failure probability associated with a lahar of intensity hi can
be estimated through the following equation:

(28)P(g(X)<0|H=hi)=Φlnhi-μβ.

The method of maximum likelihood estimation (MLE) was used for fragility
curve parameterization. This tool allows the distribution
parameters that maximize the occurrence probability of data obtained in the
Monte Carlo simulations to be determined. In this case, the objective of the MLE is to
determine the value of the bridge failure probability (pi) due to a
lahar of intensity hi that maximizes the probability of obtaining the
pairs (ni, Ni) associated with the simulations of all lahar
intensity levels hi. This is obtained by maximizing the likelihood
function, which is equal to the product of the binomial probabilities for
each height level hi.

Considering a fragility curve with cumulative log-normal distribution,
pi is replaced by the cumulative log-normal function, and parameters
μ and β are estimated. In this case, it is best to maximize the
likelihood logarithm instead of the likelihood function. Thus, parameters of
the cumulative log-normal distribution are obtained through the following
expression proposed by Lallemant et al. (2015):

μ^β^=argmaxμ,β∑i=04.0nilnΦlnhi-μβ(30)+Ni-niln1-Φlnhi-μβ.

Parameters μ and β were obtained by iterating their values and
finding the combination that maximizes Eq. (30). The process was carried out
for bridges with one span (C1) and bridges with multiple spans (C2). For
bridges without piers (C1), the result was that the likelihood function is
maximized with μ equal to 0.98 and β equal to 0.08. In this
manner, we conclude that the failure height of bridges with one span (C1)
due to lahars can be modelled with a cumulative log-normal distribution (μ=0.98; β=0.08). Regarding the bridges with two or more spans (C2),
collapse height due to lahars could be represented by a cumulative log-normal
distribution with μ equal to 0.63 and β equal to 0.13. Figure 5
shows both analytical fragility curve and parameterized fragility curve of
bridges with one span (C1) and with two or more spans (C2).

The bridge failure models presented in Sect. 4 are based on physical
models and expressions recommended in the literature; for example, this
includes the equations given by the Highway Manual of the Chilean Ministry
of Public Works (MOP, 2016) for estimating the scour supply in order to
design bridges as well as the expressions of HEC-18 (Arneson et al., 2012)
for quantifying the scour demand of the flows. All this requires an
empirical evaluation of the developed analytical failure models.

The bridge failure models are evaluated empirically using data from
historical lahars of Chile. Considering the attributes of the historical
lahars and bridges that were affected, the models quantify the net moment
(Mn) and net force (Fn) exerted by the flow on the bridge. If the
demand moment or force exceeds that of supply, the models indicate that the
analysed bridge failed due to that historical lahar. The model's result for
each bridge (failure/not failure) is compared with that indicated in the
damage reports. For the evaluation, the damage attributes and records of
lahars produced during the eruptions of the Villarrica volcano in 1964, 1971
and 2015, and the Calbuco volcano in 1961 and 2015 were used. The historical
information was compiled from Klohn (1963), Naranjo and Moreno (2004),
Moreno et al. (2006), MOP (2015a, b) and Flores (2016). The results of the bridge failure models after empirical evaluation are
shown in Table 2.

Table 2Results of the bridge failure models after empirical evaluation using
field records.

The 15 historical cases evaluated analytically with the failure models,
considering the specific inputs of the system, have the same state of damage
(failure/no failure) as that reported experimentally by the agencies. The
historical data of Table 2 consider lahars from 1.5 to 5.0 m of depth,
covering the entire range of hazard intensity of developed fragility curves
(1.5 to 4.0 m). The density of the evaluated lahars ranges from 16 000 to
19 000 N m−3, the slope from 1.0 to 1.2∘, the
bridge length from 11.3 to 72.5 m, the bridge width from 3.9 to 8.3 m,
the bridge height from 2.5 to 5.5 m, the number of deck support from 0 to
5, the bridge height from 2.5 to 8.3 m, the number of deck support from 0
to 5; the bridge materials are concrete and wood, the number of bridge lanes
are 1 and 2. Thus, the evaluated empirical data demonstrate
representativeness of the range of the basic variables of the analytical
models (Table 1).

Through this satisfactory evaluation we conclude that the existing models
integrated in the limit state functions and the values of the used variables
reflect the stability of the bridge due to a lahar flow. This allows us to
infer that the developed failure models represent the fragility of their
components in the presence of these flows.

The analysis of the models and equations used in the limit state functions
demonstrates that the lahar depth is the main variable in the quantification
of lahar loads and bridge capacity to response to these flows. The lahar
velocity, the scour demand, the hydrodynamic pressure and the height of the
debris impact depend on the flow height. Thus, it is concluded that this
variable can be used to represent the hazard intensity in the fragility
curves associated with lahars.

In order to validate parameterized fragility curves, the analytical bridge
failure probability (pa) for a lahar intensity hLahar should be
statistically compared with the empirical failure probability (pe) for
the same lahar intensity. The empirical failure probability pe can be
estimated as the proportion of bridges reached by historical lahars with
intensity hLahar that were destroyed. However, there is insufficient
empirical data to provide a statistical validation of the bridge fragility
curves. There are only 15 empirical points (hLahar, pe) to
validate two fragility curves (C1 bridges and C2 bridges). Thus, a
deficiency of empirical data on impacts of lahars on bridges is identified.

Regarding the simulations of calibrated fragility curves for the overturning
of piers and abutments, it is worth noting the greater contribution of the
moment associated with the hydrodynamic pressure than the debris impact. The
average impact moment does not exceed 0.21 % of the hydrodynamic moment
in the case of piers and 0.39 % for abutments. Moreover, it should be
noted that the contribution percentage of the impact moment decreases as the
lahar height increases.

Concerning the deck sliding, it is important to indicate that the net force
is kept relatively constant when the lahar depth is lower than or equal to 2.5 m.
This is because the tangential force of the lahar on the superstructure is
null. Afterwards, when the lahar reaches the beams and decks, the average,
minimum and maximum net forces obtained in the simulations start to
decrease. For example, the average net force is negative when the lahar
height is higher than or equal to 3.25 m, where the failure probability is 78.9 %. Moreover, if the lahar intensity is higher than or equal to 4.0 m, the
deck has a 100 % probability of sliding, because the maximum net force
obtained in the simulations is negative.

Furthermore, the results showed that the contribution of the force of the
debris impact on the superstructure is lower in relation to the hydrodynamic
force. In this particular case, the maximum average impact force represents
0.68 % of the hydrodynamic force. The reason is that the impact of debris
on the superstructure is infrequent, since it requires the height of the
impact to be higher than the height of the substructure but lower than the
road elevation. Nevertheless, if such an impact should occur, the impact force
would be high.

Regarding the fragility curves by bridge categories, the failures of bridges
from category C2 is mainly due to the overturning of piers. In fact, when
the lahar height is less than or equal to 2.0 m, the pier is the only triggering
component, because the other ones have no failure probability. The failure
probability of the abutments is greater than zero when the lahar intensity
is greater than or equal to 2.25 m. At that intensity level, the pier already has
a failure probability of 91.4 %, which means that the influence of the
abutment on the bridge failure is lower. That is why the fragility curve of
C2 bridges is similar to that of the piers overturning.

Something similar occurs in one-span bridges (C1). In this case, the
triggering component is the abutment, because it is more vulnerable to
lahars than the deck. When the flow depth is higher than 2.25 m and lower
than 2.5 m, the C1 bridges can fail only if the abutments overturn, since
the sliding probability of the deck is zero. The deck-sliding probability is
no longer null at 2.75 m but reaches a sliding probability of just 3.9 %
compared with an abutment overturning probability of 67.4 %. Therefore,
the abutment is always the main failure factor in this type of bridge.

In this paper, bridge failure models and bridge fragility curves due to
lahars are proposed, considering pier and abutment overturning as well as
deck sliding. The model development considers the calibration and
parameterization of bridge fragility curves due to lahars based on limit
state models. Two types of bridges were considered in the analysis: one-span
and multiple-span bridges. Monte Carlo simulations were applied to estimate
the failure probability given by different lahar depths. Fragility curves of
bridges were parameterized by maximum likelihood estimation, using a
cumulative log-normal distribution. Through the satisfactory empirical
evaluation of the failure models, we concluded that the models included in
the limit state functions and the proposed values to characterize lahar
flows are representative of prevailing loads and bridge capacity. The
empirical data deficiency demonstrate the need to develop more effective
protocols to report damage from volcanic events on bridges. With this, the
empirical validation of developed fragility curves is a source of future
research.

The analysis of the fragility curves demonstrated that decks fail due to
substructure overturning prior to sliding. The likelihood of deck-sliding
failure below a 2.75 m lahar height is zero because decks are above that
height. In the presence of a lahar of 2.75 m height, the pier and abutment
overturning probabilities are 98.9 % and 67.4 %, respectively. This
implies that the probability that the deck fails and the substructure does
not fail is 0.01 %, considering that these are independent events. In
addition, the research concluded that bridges with multiple spans are more
vulnerable to lahar flows compared to bridges with one span. The most
evident difference between these bridges was obtained in the lahars of
height 2.25 m. Given this intensity, bridges with one span (C1) have a 0.3 % probability of failure, while those with multiple spans (C2) have a
92.0 % probability of failure. This result was expected because, when
impacted by lahars, piers are more susceptible to overturn than abutments.

With the developed fragility curves, agencies can determine the failure
probability of bridges due to a lahar presenting a specific depth. The
proposed failure models can be adapted and calibrated to bridge designs that
are different from the structures accounted for in the article. When
required, the supply function considered in the models can be conditioned to
local bridge design standards and adjusted accordingly.

For the application of these models, it is recommended that expected hazard
scenarios, in terms of recurrence and intensity, should be first simulated.
The resulting hazard intensity can then be estimated for the affected road
network, in particular exposed bridges, and their failure probability can
consequently be calculated. Further research is being conducted in this
regard;
that is a computational platform is being developed for the consistent
application of the developed fragility curves for the exposed networks. With
this, local authorities can review their road and bridge designs and
existing infrastructure in order to assess and apply mitigation strategies
prior to the occurrence of a volcanic event.

The data used
in the failure models are available in AASHTO (2012), Arneson et al. (2012)
and MOP (2016). Other details of the analytical scour models are provided in
NCHRP (2010) and can be provided on demand.

The authors thank the National Commission for Scientific and Technological
Research (CONICYT), which has financed the FONDEF Project ID14I10309
Research and Development of Models to Quantify and Mitigate the Risk of
Natural Hazards in the National Road Network. Likewise, we express our
gratitude to the institutions that participated and contributed to this
research project, especially to the Research Center for Integrated Disaster
Risk Management CONICYT/FONDAP/15110017 (CIGIDEN), the Highways Department
of the Chilean Ministry of Public Works and the National Geology and Mining
Service of the Chilean Ministry of Mining, the National Emergency Office of
the Ministry of Interior and Public Security (ONEMI) and the Chilean
Association of Concessionaires (COPSA).

Zuccaro, G. and De Gregorio, D.: Time and space dependency in impact damage
evaluation of a sub-Plinian eruption at Mount Vesuvius, Nat. Hazards, 68,
1399–1423, 2013.

Search articles

Download

Short summary

One of the main volcanic processes affecting road bridges are lahars, which are flows of water and volcanic material running down the slopes of a volcano. In this paper, bridge failure models due to lahars are proposed and, based on these, fragility curves are developed. Fragility curves are parameterized by maximum likelihood estimation, assuming a cumulative log-normal distribution. Bridge failure models are empirically evaluated using data of 15 bridges that were affected by lahars.

One of the main volcanic processes affecting road bridges are lahars, which are flows of water...